Method and apparatus for efficient routing of variable traffic

ABSTRACT

A method and apparatus for provide highly efficient traffic routing for a wide range of possible traffic matrices (TM) in an intra-domain network. That routing optimally balances the traffic loads over a range of traffic matrices so as to minimize the deviation for any particular traffic matrix from the optimal routing. Such a routing provides a guaranteed performance ratio against the best possible network routing. The invention utilizes a method of optimally configuring a traffic network based on solving a linear program to obtain the optimal routing, and then configuring the routing on the network accordingly.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.10/797,762 filed Mar. 10, 2004, entitled, “METHOD AND APPARATUS FOREFFICIENT ROUTING OF VARIABLE TRAFFIC”, (now U.S. Pat. No. 7,388,842)which claims the benefit of U.S. Provisional Patent Application No.60/455,462, filed Mar. 13, 2003. All of the above-cited applications areherein incorporated by reference in their entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to intra-domain traffic engineering. Morespecifically, embodiments of the present invention relate to improvedmethods and systems for intra-net traffic routing.

2. Description of the Related Art

Because of the increasing cost and complexity of large operationalinternet protocol (IP) networks, traffic engineering has becomeimportant in recent years. Good traffic engineering can significantlyimprove the management and performance of operational IP networks whilereducing costs. Prior art traffic engineering was based on a solidunderstanding of traffic flows when designing and configuring trafficrouting protocols.

It has been widely accepted in the traffic engineering community that agood understanding of the traffic matrix (TM) and the dynamics oftraffic flows can lead to better utilization of link capacities throughbetter traffic routing. Theoretically, if the TM is exactly known, thenan optimal traffic routing can be obtained by solving the correspondingmulti-commodity flow problem instance, reference D. Mitra and K. G.Ramakrishna, “A Case Study of Multiservice, Multipriority TrafficEngineering Design for Data Networks,” Proceedings of IEEE GLOBECOM,pages 1077-1083, IEEE 1999. Based on the TM it is possible to establishlink weights, that being the percentage of traffic routed through aparticular link in a network of link and nodes between anorigin-destination pair, that can be tuned to yield near-optimalutilization.

Unfortunately, measuring and predicting traffic demands are illusiveproblems. Flow measurements are rarely available on all links andEgress/Ingress points of the network. It is even harder to obtain a goodpicture of Origin-Destination (OD) flow aggregates. Moreover, trafficdemands change over time, both in the short term and in the long term,and are subject to special events or failures, either internal orexternal to the network. Despite recently developed models andmeasurement tools that enable extrapolation and estimation of trafficdemands, it appeared that the best to hope for was an approximatepicture of demand, and not necessarily a very good or very current one.

Even if current traffic demands are known, their dynamic nature poses achallenge: while it is desirable to modify the routing to be highlyefficient for the current traffic demands, modifying the traffic routingcan potentially cause disruptions in service due to path changes andconvergence times while the system reaches a consistent state.

This basic premise, however, never seems to have been quantified: justhow important is accurate knowledge of traffic demands to obtaining goodnetwork utilization? Since traffic demands are dynamic and illusive, anoptimal routing solution at one time could be a poor routing solution atanother, and adjusting the routing presents multiple issues. That posesa question: is it possible, possibly better, to obtain a robust routingthat guarantees a nearly optimal utilization based on only a fairlylimited knowledge of the applicable traffic demands?

Good traffic engineering would seem to call for a design that is robustunder a wide range of conditions. That is, a routing that can performnearly optimally for a wide range of traffic demands. Thus methods ofand systems for routing traffic in a nearly optimal way over a widerange of traffic demands and that handles link failures well would behighly useful.

SUMMARY OF THE INVENTION

Embodiments of the present invention provide highly efficient trafficrouting for a wide range of possible traffic matrices (TM) in anintra-domain network. That routing optimally balances the traffic loadsover a range of TMs so as to minimize the deviation for any particularTM from the optimal routing. Such a routing provides a guaranteedperformance ratio against the best possible network routing.

Embodiments of the present invention utilize a method of optimallyconfiguring a traffic network based on solving a linear program toobtain the optimal routing, and then configuring the routing on thenetwork accordingly.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above recited features of the presentinvention can be understood in detail, a more particular description ofthe invention, briefly summarized above, may be had by reference toembodiments, some of which are illustrated in the appended drawings. Itis to be noted, however, that the appended drawings illustrate onlytypical embodiments of this invention and are therefore not to beconsidered limiting of its scope, for the invention may admit to otherequally effective embodiments.

FIG. 1 illustrates an intra-net system showing both nodes and links;

FIG. 2 illustrates a routing scheme for an origin to a destination;

FIG. 3 illustrates a flow diagram of a method of configuring theintra-net system shown in FIG. 1;

FIG. 4 illustrates a C₆ network topology; and

FIG. 5 illustrates a node that is in accord with the principles of thepresent invention.

To facilitate understanding, identical reference numerals have beenused, wherever possible, to designate identical elements that are commonto the figures.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Prior to the present invention it was not known how to design anintra-net IP traffic routing that optimally balanced a wide range ofpossible traffic demands so that a good, nearly optimal, traffic routingwas obtained under a wide range of traffic demands. The principles ofthe present invention provide for methods of configuring a network toprovide a traffic routing that balances a wide range of possibleintra-net IP traffic demands so that a good, nearly optimal, trafficrouting is obtained under a wide range of traffic demands.

Embodiments of the present invention can have a low sensitivity ofrouting quality to traffic demands. This insensitivity has previouslynot been well understood or precisely quantized: The principles of thepresent invention address the issue of just how well an intra-net IPtraffic routing that is designed with either no knowledge, or with onlya limited knowledge, can perform. Systems that are in accord with thepresent invention can have guaranteed performance over widely varyingtraffic demand. Embodiments of the present invention are capable ofefficient handling of link failures without network re-configuring.

Figure illustrates a typical intra-net system 100 that is comprised ofnodes, labeled N1-N2, and links, L1-L34, that interconnect the nodes.The nodes represent communication points, be they individual computersor servers, while the links represent the communication paths to andfrom the individual links. The links can be wire, cable, optical fiber,microwave, or other techniques of communicating information.

In what follows, a routing specifies how traffic between eachOrigin-Destination (OD) pair is routed through the links across anetwork. In FIG. 1, node N7 is designated as the origin while node N15is designated as the destination. Thus, node N7, say a communication netin Chicago, wants to send traffic to node N15, say a communication netin Los Angeles. Typically there is path diversity, that is, there aremultiple paths for each OD pair, and each path routes a fraction of thetraffic. This is illustrated in FIG. 2. Assuming the origin N7 wants tosend information to node N15, and assuming just two additional nodes 202and 204 exist, then traffic can be routed numerous ways along the206-216. However, FIG. 2 shows ½ the traffic flowing through link 216, ¼flowing through link 206, and ¼ flowing through link 212. Furthermore,FIG. 2 shows that the ¼ flowing through link 206 also flows through link210 and the ¼ flowing through link 212 flows through link 214. Finally,none of the traffic from node N7 to node N15 flows through link 208.Various other paths exist (for example, link 206 to link 208 to link214) and other link weights (traffic percentages) can be assigned to thevarious paths.

The Open Shortest Path First (OSPF) and Intermediate System-IntermediateSystem (IS-IS) are protocols that specify a routing through a set oflink weights. The traffic between each OD pair is always routed onshortest path(s) between the origin and destination (with respect tothese weights). Typically, there are multiple shortest paths. When thishappens, each router splits the outgoing traffic evenly on allapplicable interfaces. By controlling the link weights, many routingsare possible.

One previously recommended link weight default setting is to use linkweights that are inversely proportional to the link capacities. Withmore fine-tuned traffic engineering it is typically possible to selectlink weights that are expected to work well on the projected TM. TheOSPF routing is the routing obtained by the estimated link weights inaccord with available data. The OSPF routing should match reasonablyclosely the actual routing used by many ISPs. The MPLS protocol allowsfor a rich specification of routings and more fine tuned trafficengineering.

The principles of the present invention provide for a method ofoperating a traffic network using an optimal routing for variable oruncertain traffic demands. Those principles further provide for anetwork that routes traffic in accord with constraints on OD pairdemands.

The method uses a modeled variability or uncertainty for eachOrigin-Destination (OD) pair to specifying a range of possible demands.The method determines the “best possible” routing for any traffic demandwithin the specified ranges (that is, for each Origin-Destination pair,the demand is within the range specified for that pair). The notion of“best” is based on a competitive ratio. A routing's competitive ratio isthe maximum over all traffic demands of the routing's congestion to theminimum possible congestion. An optimum routing is the one which has thesmallest competitive ratio. The method determines the best possiblerouting by resolving a linear program that minimizes the competitiveratio subject to multiple constraints. The method simultaneouslydetermines the optimal routing and certifies that the competitive ratiois satisfied.

The relevant characterization of each routing is what fraction of thetraffic for each OD pair is routed along each link. Thus, the trafficrouting is specified by a set of values f_(ab)(i,j) that determine thefraction of traffic demand from a to b that is routed on a link (i,j).Note that the values f_(ab)(i,j) for a given OD pair a→b specify a flowof value 1 from a to b. When the routing routes a demand d_(ab) for theOD pair a→b the contribution of this demand to the flow on a link (i,j).is d_(ab) f_(ab)(i,j).

A common metric for the performance of a given routing with respect to acertain TM is the maximum link utilization. This is the maximum, overall links, of the total flow on a link divided by the capacity of thelink. Formally, the maximum link utilization of a routing f on TM D(where d_(ab) is the demand from a to b) is:

Max Σ d_(ab)f_(ab)(i,j)±cap_(ij)

(i,j) εlinks a,b

where cap_(ij) is the capacity of the link (i,j).

An optimal routing for a certain TM D is a routing which minimizes themaximum link utilization. Formally, the optimal utilization for a TM Dis given by

${{OPTU}(D)} = {\min\limits_{f❘{f\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{routing}}}{\max\limits_{{({i,j})} \in \;{links}}\frac{\sum\limits_{a,b}{d_{ab}{f_{ab}\left( {i,j} \right)}}}{{cap}_{ij}}}}$

The performance ratio of a given routing f on a given TM D measures howfar f is from being optimal on the TM D. It is defined as the maximumlink utilization of f on D divided by the minimum possible maximum linkutilization on this TM. Formally,

${{PERF}\left( {f,\left\{ D \right\}} \right)} = \frac{{\max\left( {i,j} \right)} \in {{links}\;{\sum\limits_{a,b}\;{d_{ab}{{f_{ab}\left( {i,j} \right)}/{cap}_{ij}}}}}}{{OPTU}(D)}$

The performance ratio is always at least 1; it is exactly 1 if and onlyif the routing is optimal for D.

It is well known that the optimal routing for a given TM can be computedby solving a corresponding multi-commodity flow linear program. Notethat this routing is optimized for a specific TM, thus, it does notprovide performance guarantees for other TMs. This is important, since,as mentioned earlier, traffic patterns change over time and it is alsonot generally possible to obtain a good estimate of the current TM.

The definition of the performance ratio follows the competitive analysisframework where performance guarantees of a certain solution areprovided relative to the best possible solution. The definition ofperformance ratio of a routing can be extended to be with respect to aset of TMs. Let D be a set of TMs. The performance ratio of a routing fon D is defined as

$\begin{matrix}{{{PERF}\left( {f,\left\{ D \right\}} \right)} = {\max\;{{PERF}\left( {f,\left\{ D \right\}} \right)}}} \\{D \in D}\end{matrix}.$

A routing f is optimal for D if and only if it minimizes the performanceratio, that is PERF(f,D) is minimal. The performance ratio PERF(f,D) isalways at least 1, but the best possible performance ratio on the set ofTMs D can be strictly larger than 1; since generally, a single routingthat is optimal for all TMs in the set may not exist.

When the set D includes all possible TMs, the performance ratio isreferred to as the oblivious performance ratio of a routing. Theoblivious ratio is the worst performance ratio a routing can obtain withrespect to any TM. A routing with a minimum oblivious ratio is anoptimal oblivious routing, and its oblivious ratio is the optimaloblivious ratio of the network.

Note that the performance ratio constitutes a comparative measure ofdifferent routings, for a given topology and set of TMs. It is not,however, a meaningful comparative measure between different networktopologies, it is defined relative to the minimum possible maximum linkutilization, but the min max utilization itself varies with topology.Also there can be many possible optimal routings and they can differ inhow they perform on specific TMs.

Known tools allowed for optimizing the routing with respect to a givenTM, but beyond specific highly structured topologies (such as hypercubicnetworks), not much was known about constructing an optimal routing withrespect to a rich set of demands and what performance ratios could beexpected. Existentially, an upper bound exists: all symmetric networks(that is, networks where link capacities are the same in bothdirections, as is typically the case with large backbone networks) havea routing with an oblivious ratio that is at most polylogarithmic in thenumber of nodes. Further, an optimal oblivious routing can be computedin polynomial time. The polynomial time algorithm is based on theEllipsoid algorithm and as such does not provide a practical way toproduce the routing for large networks.

However, the principles of the present invention provide for enhanced LP(linear program) models (see details subsequently) that are comprised ofa single polynomial size LP formulation. Those models can be extended tooptimize routings with respect to range restrictions on OD-pair demands.(a restricted set of TMs will provide a better ratio). The inventorshave simulated and solved the LP models using the CPLEX LP solver, alarge-scale mathematical programming software (2003 version), seehttp://www.cplex.com. However, other LP solvers could be applied equallyas well).

FIG. 3 is a flow chart of a method 300 of setting up and operating anintra-net that is in accord with the present invention. That methodstarts at step 302 and proceeds at step 304 with obtaining basicinformation about the intra-net, specifically the nodes and the links.Then, at step 306 the method determines the possible origin-destinationpairs. Not all nodes need be part of origins or destinations. Then, atstep 308, the method proceeds by obtaining the optimal routing and ratioof the network. Then, at step 310, the optimal routing is used toconfigure the network. Finally, with the network set-up, the methodstops at 312.

The LP (linear program) models are based on a set of lemmas that areused to reduce the size of the models. The lemmas lead to a result thatestablishes that an optimal oblivious routing (and oblivious ratio) of anetwork can be computed in polynomial time. That result leads tosimplified LP models that allow for considerably faster running times.

Basic lemmas show that for the purposes of computing a performance ratioof sections of a network where path diversity is not possible (allrouting would perform the same) can be “factored out” of the networktopology. This reduces the size of the input topologies.

A first lemma is that removal of degree-one nodes does not affect theoblivious ratio of the network. Similarly, such removal does not affectthe optimal performance ratio with respect to any set of TMs.

The first lemma is a corollary of a second lemma: the optimal obliviousratio of a network can be computed by partitioning the network to 2-edgeconnected components and taking the maximum of the oblivious ratio overthese components.

Proof: If the network G is not 2-edge connected, it can be partitionedinto two non-empty components A and B that are connected by an edge(a,b) where a ε A and b ε B. It is easy to see that the optimaloblivious ratio of G is at least that of the maximum optimal obliviousratio of A and B: The optimal performance ratio obtained on G for TMsthat have positive demands only at OD pairs that both lie in A(respectively, both lie in B) is equal to the optimal oblivious ratio ofA (respectively, B). To see that, observe that all flow leaving/enteringA must go through the edge (a. b), thus there is never an advantage toroute demand internal to A through the edge (a, b) and out of A, sincethis flow will have to traverse back on the same edge and form a flowcycle (the symmetric argument holds for B). The optimal oblivious ratioof G is at least the optimal performance ratio on these more restrictedset of TMs.

We now argue the converse, that the optimal oblivious ratio on G is atmost the maximum optimal oblivious ratio of A and B. Let f_(A)(respectively, f_(B)) be an optimal oblivious routing on A(respectively, B). We extend the routings f_(A) and f_(B) to a routingf_(G) on G as follows:

all OD pairs internal to A or B are routed according to the respectiverouting. The routing for OD pair (a′. b′) where a′ A and b′ B (similarconstruction for pairs (b′, a′)) is routed by concatenating the routingf_(A) from a′ to a with a flow of value 1 from a to b with the routingf_(B) from b to b′. Consider now a TM D_(G) on G. We will show that theperformance ratio of f_(G) on G is at most the maximum optimal obliviousratio of A and B. We can assume (by scaling D_(G)) that the maximum edgeutilization of the optimal routing of D_(G) is 1. Thus, the performanceratio of f_(G) on D_(G) is equal to the maximum edge utilization. We nowdefine the TMs D_(A) and D_(B) for A and B, respectively; where D_(A) isobtained by aggregating all the demand into/from nodes in A from/intonodes in B to demands from/into the node a (D_(B) is similarly defined).

The maximum edge utilization of f_(G) on D_(G) is the maximumutilization over the edges of A, the edges of B, and the edge (a, b).The utilization of the edge (a, b) is equal to the aggregated demandbetween A and B. Since the utilization must be at least that also forthe optimal routing for D_(G), from our scaling assumption It followsthat the aggregated demand is at most the capacity of (a, b), and thusthe utilization is at most 1. The maximum edge utilization over theedges of A is equal to the utilization of f_(A) on the demands D_(A),which is at most the optimal oblivious ratio of A (similar for B.) Thesymmetric argument for the edges of B concludes the proof.

A network with symmetric directed links (that is, link capacities areequal in both directions) always has an optimal oblivious routing thatis symmetric. Since links are often symmetric, we use this to reduce thesize of our LP models.

Based on the foregoing a linear program model can be developed. It hasbeen shown that an optimal oblivious routing can be computed by solvinga Linear Program (LP) with a polynomial number of variables, butinfinitely many constraints (every possible TM has a set ofconstraints). We refer to this LP as the “master LP.” We use thefollowing notation: the term “link” for an undirected edge, “edge” for adirected edge, and let link-of (e) be the link corresponding to edge e.We use the notation:

${f_{ij}(l)} = {\underset{{e\text{:}{link}\text{-}{{of}{(e)}}} = l}{\mspace{40mu}{\sum f_{ij}}}\;{(e).}}$

The master LP model is:

$\begin{matrix}{{\min\; r}{{f_{ij}(e)}\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{routing}}\mspace{11mu}{{\forall\mspace{14mu}{{links}\mspace{14mu} l\mspace{14mu}{\forall{{TMs}\mspace{14mu} D\mspace{14mu}{with}\mspace{14mu}{{OPTU}(D)}}}}} = {1:\mspace{11mu}\mspace{79mu}{{\sum\limits_{ij}\;{{f_{ij}(l)}d_{ij}}} \leq r}}}} & (1)\end{matrix}$

Furthermore, given a routing f_(ij)(e), the constraints (1) can betested by solving, for each link l, the following “slave LP,” andtesting if the objective is ≦r or not.

$\begin{matrix}{{\max\mspace{14mu}{\sum\limits_{ij}\;{{f_{ij}(l)}{d_{ij}/{{cap}(l)}}}}}{{g_{ij}(e)}\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{flow}\mspace{14mu}{of}\mspace{14mu}{demands}\mspace{14mu} d_{ij}}{\forall\mspace{14mu}{{{links}\mspace{14mu} m\text{:}{\sum\limits_{ij}{g_{ij}(m)}}} \leq {{cap}(m)}}}{\forall\mspace{14mu}\left. {{demands}\mspace{14mu} i}\rightarrow{j:\mspace{14mu}{d_{ij} \geq 0}} \right.}} & (2)\end{matrix}$

Thus, the LPs (2) can be used as a separation oracle for the constraints(1), giving polynomial solvability using the Ellipsoid algorithm.

A simpler LP model that enables us to efficiently process largernetworks can be derived. For simplicity, our discussion focuses oncomputing the optimal oblivious routing, that is, a routing thatprovides performance guarantees with respect to all possible TMs. Wethen state the generalized LP model we used to support intervalrestrictions on OD pairs demands.

The first simplification one might try to apply is to somehow directlycombine the master and slave LPs, to yield a single polynomial size LPinstance. However, there are two obstacles: first, both f_(ij)(l) andd_(ij) would be variables in a combination, resulting in quadratic (nonlinear) constraints, and second, requiring that a maximum over an LP be≧r is not readily modeled. Fortunately, the LP dual of the systems (2)leads to a nice characterization of good oblivious routings:

Theorem 1: A routing f_(ij)(e) has oblivious ratio ≧r if and only ifthere exist weights π(l, m) for every pair of links l, m such that:

P1Σ_(m)cap(m)π(l, m)≦r for every link l.

-   -   P2. For every link l, for every demand i→j, and for every path        h₁, h₂, . . . , h_(p), from i to j,

${{f_{ij}(l)} \leq {{{cap}(l)}{\sum\limits_{k = 1}^{p}\;{{{\pi\left( {l,{{link}\text{-}{{of}\left( h_{k} \right)}}} \right)}.P}\; 3\mspace{59mu}\pi\mspace{14mu}\left( {l,m} \right)}}} \geq {0\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{links}\mspace{14mu} l}},m$

Proof: The proof is essentially duality applied to the slave problem.Requirements (P1)-(P3) are equivalent to stating that the slave LP'shave dual objective values≧r.

(“if” direction): Let f_(ij)(e) be a routing, and π(l,m) be weightssatisfying requirements (P1)-(P3). Suppose (g,d) is a flow of demands dwith maximum utilization of 1, and let l be a link. For each demand i→j,g_(ij), must contain paths from i→j of total weight d_(ij). From (P2)and (P3), summing over all paths, we have

${{f_{ij}(l)}d_{ij}} \geq {{cap}\left( {l{\sum\limits_{h}\;{{\pi\left( {l,{{ling}\text{-}{{of}(h)}}} \right)}{g_{ij}(h)}}}} \right.}$

Summing over all demands i→j, we have

${{\sum\limits_{ij}\;{{f_{ij}(l)}d_{ij}}} \leq {{{cap}(l)}{\sum\limits_{ij}{\sum\limits_{h}\;{{\pi\left( {l,{{link}\text{-}{{of}(h)}}} \right)}{g_{ij}(h)}}}}}} = {{{{cap}(l)}{\sum\limits_{m}\left( {{\pi\left( {l,m} \right)}{\sum\limits_{ij}{g_{ij}(m)}}} \right)}} \leq {{{cap}(l)}{\sum\limits_{m}\left( {{\pi\left( {l,m} \right)}{{cap}(m)}} \right.}}}$

The last inequality follows since g fits within the edge capacities

$\mspace{31mu}\begin{matrix}{\left( {{\sum\limits_{ij}{g_{ij}(m)}} \leq {{{cap}(m)}{{cap}(m)}}} \right),\mspace{14mu}{{and}\mspace{14mu}{from}\mspace{14mu}\left( {P\; 1} \right)}} \\{{\sum\limits_{ij}\;{{f_{ij}(l)}d_{ij}}} \leq {{{cap}(l)}{\sum\limits_{m}{{\pi\left( {l,m} \right)}{{cap}(m)}}}} \leq {{{cap}(l)}{r \cdot}}}\end{matrix}$

This says that for any demand d which can be routed with congestion 1,f's utilization on any link l is at most r, which is what we wanted.

(“only if” direction): Let flow f_(ij)(e) have oblivious ratio≦r, andlet l be a link. The dual of the slave LP (2) for link l is:

$\begin{matrix}{{\min{\sum\limits_{m}\;{{{cap}(m)}{\pi\left( {l,m} \right)}}}}{\forall\mspace{14mu}\left. {{demands}\mspace{14mu} i}\rightarrow{j:\mspace{14mu}{{\lambda_{ij}\left( {l,j} \right)} \geq {\int_{ij}{(l)/{{cap}(l)}}}}} \right.}} & (3) \\{{{\forall\mspace{14mu}\left. {{demands}\mspace{14mu} i}\rightarrow j \right.},\mspace{14mu}{{\forall\mspace{14mu}{{edges}\mspace{14mu} e}} = \left. i^{\prime}\rightarrow{j^{\prime}:{{{\pi\left( {l,{{link}\text{-}{{of}(e)}}} \right)} + {\lambda_{ij}\left( {l,i^{\prime}} \right)} - {\lambda_{ij}\left( {l,j^{\prime}} \right)}} \geq 0}} \right.}}{\forall\mspace{14mu}{{links}\mspace{14mu}{m:{{\pi\left( {l,m} \right)} \geq 0}}}}{{\forall\mspace{14mu}\left. {{demands}\mspace{14mu} i}\rightarrow j \right.},{\forall\mspace{14mu}{{nodes}\mspace{14mu}{k:\mspace{14mu}{{\lambda_{ij}\left( {l,k} \right)} \geq 0}}}}}{{\forall\mspace{14mu}\left. {{demands}\mspace{14mu} i}\rightarrow{j:\mspace{14mu}{\lambda_{ij}\left( {l,i} \right)}} \right.} = 0}} & (4)\end{matrix}$

The variable λ_(ij)(l,k) is the dual multiplier on the flow conservationconstraint for demand i→j at node k. Since there is no flow conservationconstraint in the primal at node i, we have introduced λ_(ij)(l,i),fixed at 0, for convenience. The variable π(l, m) is the dual multiplieron the capacity constraint for link m.

Since f_(ij)(e) has oblivious ratio≦r, the primal slave LP for any linkl must have optimum≦r, and hence also the dual slave LP for link l musthave optimum≦r. Hence, the ππ(l,m) from the dual slave LPs satisfy (P1).Trivially, they also satisfy (P3). Now, let i→j be a demand, and h₁, . .. , h_(p) be a path from l to j. Summing up constraint (4) over edgesh₁, . . . , h_(p) we have

${{\sum\limits_{k = 1}^{p}\;{\pi\left( {l,{{linki}\text{-}{{off}\left( h_{k} \right)}}} \right)}} + {\lambda_{ij}\left( {l,i} \right)} - {\lambda_{ij}\left( {l,j} \right)}} \leq 0$

Since λ_(ij) (l,i)=0

${\sum\limits_{k = 1}^{p}\;{\pi\left( {l,{{linki}\text{-}{{off}\left( h_{k} \right)}}} \right)}} \leq {\lambda_{ij}({lji})} \leq {{f_{ij}(l)}/{{cap}(l)}}$

so the π(l, m) satisfy (P2).

We next apply Theorem 1 to show that the problem can be solved by asingle polynomial-sized LP. This results in a significant algorithmicperformance gain, since it means the problem can be solved by the moreefficient interior-Point algorithm.

Theorem 2: The oblivious ratio of a network can be computed by a singleLP with O(mn²) variables and O(nm²) constraints.

Proof. We introduce the variables p_(l)(i, j), for each link I and ODpair i, j. The variable p_(l)(i, j) is the length of the shortest pathfrom i to j according to the link weights π(l, m) (for all m). Theintroduction of these variables allows us to replace the exponentialnumber of constraints (for all possible paths) in Requirement (P2) ofTheorem 1 with a small polynomial number of constraints.

$\left. \left. {{{mm}\mspace{14mu} r}{{f_{ij}(e)}\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{routing}}{\forall\mspace{14mu}{{links}\mspace{14mu}{l:{{\sum\limits_{m}\;{{{cap}(m)}{\pi\left( {l,m} \right)}}} < r}}}}} \right) \right)$∀  links  l, ∀  pairs  i → j:  f_(ij)(l)/cap(l) − s_(l)⁺(l, j) + s_(l)⁻(l, j) = p_(l)(l, j)∀  links  l, ∀  nodes  l, ∀  edges  e = j → k:   π(l, link-of(e)) + p_(l)(i, j) − p_(l)(i, k) ≤ 0∀  links  l, m:  π(l, m) ≤ 0 ∀  links  l, ∀  nodes  i:  p_(l)(i, i) = 0∀  links  l, ∀  nodes  i.j_ ≤ 0

This LP has 0(mn²)constraints.

To compute the oblivious ration when demand i→j is restricted to therange [a_(ij),b_(ij)], we modify the slave LP (2) by replacing theconstraint d_(ij)≦b_(ij), and following that change through the dual LP(3) into the single LP (5). This results in the introduction of theslack variables s_(l) ⁻(i,j) and s_(l) ⁺(i,j,) for the lower and upperbound constraints on d_(ij).

$\left. \left. {{{mim}\mspace{14mu} r}{{f_{ij}(e)}\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{routing}}{\forall\mspace{14mu}{{links}\mspace{14mu}{l:\mspace{14mu}{{\sum\limits_{m}\;{{{cap}(m)}{\pi\left( {l,m} \right)}}} < r}}}}} \right) \right)$∀  links, l  ∀  pairs  i → j : f_(ij)(l)/cap(l) − s_(l)⁺(l, j) + s_(l)⁻(l, j) = p_(l)(l, j)∀  links  l, ∀  nodes  l, ∀  edges  e = j → k:   π(l, link-of(e)) + p_(l)(i, j) − p_(l)(i, k) ≤ 0∀  links  l, m:  π(l, m) ≤ 0 ∀  links  l${\sum\limits_{ij}\;\left( {b_{ij}{s_{l}^{-}\left( {i,j} \right)}} \right)} \leq 0$∀  links  l, m:  π(l, m) ≤ 0 ∀  links  l, ∀  nodes  i:  p_(l)(i, i) = 0∀  links  l, ∀  nodes  i, j:  p_(l)(i, j) ≥ 0∀  links  l, ∀  nodes  i, j:  s_(l)⁻(i, j) ≥ 0∀  links  l, ∀  nodes  i, j:  s_(l)⁺(i, j) ≥ 0

This reduces to the single LP if the bounds are [0∞).

We analyze the optimal oblivious ratio for two simple families ofnetwork topologies: The cycle topology C_(n), has n nodes that areconnected in a cycle pattern with unit capacity links. The cliquetopology Kn has n nodes connected via a complete graph, that is, thereis a unit capacity edge connecting any two nodes.

Our interest in these networks is two-fold. First, these networks admitan optimal oblivious ratio bounded by 2 (even for large values of n).This provides some indication that a small “constant” optimal obliviousratio is possible as ISP networks scale up. Second, these simpletopologies better illustrate our metrics and the notion of a good“demand oblivious” routing.

These two families of topologies are highly homomorphic (“look the same”from any node). We will find the following lemma useful for analyzingthem:

Lemma 7.1: If two nodes u and v are homomorphic under some homomorphismH, then there exists an optimal oblivious routing such that the routingfrom u to a node w on an edge e is equal to the routing from v=H(u) toH(w) on the edge H(e).

FIG. 4( a) illustrates the C₆ topology. Consider first a TM thatconstitutes of a positive demand on the single OD pair 0→1. The optimalrouting for this TM (that is, the routing that minimizes the maximumutilization) balances the load on the two disjoint paths form 0 to l:half the demand flows on the direct edge (0, 1) and the other half onthe 5-edge path (0,5,4,3,2, 1) (this “even-split” routing is illustratedin FIG. 4( b). The shortest-path routing, which sends the flow of eachOD pair on the shorter of the two available paths (that is, for our TMit would send all flow on the direct edge (0, 1)), has performance ratioof 2 on our TM, as the maximum link utilization obtained by this routingis double that of the even-split routing.

We next consider another simple TM where there are unit demands on all“consecutive” OD pairs i→(i+1) mod 6 (for i=0, . . . , 5), and no demandon other pairs. We first consider routing the demand of each OD pairevenly on the two available paths (using the “even-split” routing weused in part (b)). The flow routes of the routings on this TM areillustrated in FIG. 4( c): The solid lines indicate the routes used bythe shortest-path routing. The solid and dotted paths together are usedby the even-split routing, which induces a flow of half from every ODpair demand on every edge. We thus obtain that the even-split routinghas link utilization of 3. In contrast, the shortest-path routing (solidlines in (c)) would send on each edge only the demand due to the verysame OD pair, resulting in maximum link utilization of 1. Thus, theperformance ratio of the even-split routing on this TM is at least 3 (infact, it is exactly 3 since the direct routing is optimal). The sameargumentation can be carried over to other cycle topologies C n it isnot hard to see that the even-split routing would have utilization ofm/2 and performance ratio of n/2 (since the shortest-path routing hasmaximum utilization of 1).

What this means is that the even-split routing is a bad obliviousrouting. The next issue is finding a good oblivious routing. We considergeneral TMs and argue that the shortest-path routing has oblivious ratioof 2. The shortest-path routing for all OD pairs that utilize the edge(0, 1) is illustrated in FIG. 4( d). Consider an arbitrary TM, and theedge with highest utilization according to the shortest-path routing.Without loss of generality we can assume that this edge is (0, 1). Werefer to the edge (3, 4) as the “opposite edge” from (0, 1). (In generalfor even valued n, the opposite edge of (i,(+1)mod n) on C_(n), is theedge (i+n/2, (i+1+n/2)mod n).) It is not hard to verify the followingproperty: for every OD pair that its demand is routed by theshortest-path routing on the edge (0, 1), the two edges (0, 1) and itsopposite edge (3, 4) “disconnect” the pair (the two edges are “cut”edges). What this means is that for any routing, the sum of flows thatare due to demand 0→1 on the edge (0, 1) and its opposite edge must beat least the demand between 0 and 1. It follows that for any routing,the maximum utilization over the two edges (0, 1) and its opposite mustbe at least half of the utilization of (0, 1) under shortest-pathrouting. We thus obtain that the performance ratio of the shortest-pathrouting is at most 2.

We next provide a formal proof that states that the best possibleperformance ratio for the cycle C_(n) is 2-2/n. We shall see that theoptimal oblivious routing will send some flow along the longer path (butmost flow along the shorter path). We use the following notation: wenumber the nodes of C_(n), as 0, . . . , n−1, with node numbers takenmodulo n, so that the edges are (i,i+1) and (i+1,i)·f_(a,b)(i, j)) isthe oblivious routing's flow for demand a a→b on edge i→j.

Lemma 7.2: The optimal oblivious performance ratio for C_(n), (the cycleon n vertices) is 2-2/n.

Proof: We first show that the ratio is at least 2-2/n: From symmetry(see Lemma 7.1) and flow conservation considerations, there is anoptimal oblivious routing of the following form: for some x_(i)≧0

fa, a + k(a + i, a + i + 1) = x_(k)  ∀a, k ∈ [1, n − 1].∀i ∈ [0, k − 1]fa, s + k(a − i, a − i) + 1x_(k)   ∀a, k ∈ [1, n − 1] f_(a, b)(c, d) = 0

-   -   for all other a,b,c,d

x_(k) = 1 − x_(n − k)  ∀k ∈ [1, n − 1]

For any a, a demand a→(a+1) of size 2 can be routed within unitcapacities, so from the load on edge (a, a+1), we have that the optimalratio is at least2*fa,a+i(a,a+1)=2x _(i)

On the other hand, a demand for all a→a+1(a=0 . . . n−1) of size 1 canalso be routed within unit capacities, so from the load on a particularedge:

-   (a,a+1), we have that the ratio is at least    ˜Σfa _(—) i,a+i1(a,a+1)=x ₁+(n−1)(1−x ₁)i=0 . . . n−1

From the above two bounds we obtain that the optimal ratio is at leastmax{2x ₁ ,x ₁+(n−1)(1−x ₁)}≧_(—)2-2/n

-   -   (the maximum is minimized when x₁=(n−1/n).

It remains to show that the optimal ratio is at most 2-2/n. Consider therouting obtained by setting x_(k)=(n−k)/n, we show that this routing hasoblivious performance ratio of at most 2-2/n. Consider, without loss ofgenerality, the edge (0, n−1). A demand da,b, with 0≦a<b≦n−1, musteither be routed using the edge (0, n−1), or be routed on the path (a,a+1. . . , b), using b−a edges. Similarly, a demand d_(b,a) with0≦a<b≦n−1 must either be routed using the edge (n−1, 0) or be routed onthe path (b, b−1, . . . , a) using b−a edges. Consider now a TM that canbe routed such that each edge has at most 1 unit of flow on it. Itsuffices to show that our routing has utilization at most 1-2/n on thatTM. Consider such demands and supposed that the optimal routing for thatTM is such that none of the demands were routed on the edge (0, n−1).Then the total edge load generated on the edges (0,1, . . . , n−1) wouldbe:

$\left. {\sum\limits_{0 \leq a < b \leq {n - 1}}{\left( {b - a} \right)\left( d_{b,a} \right)}} \right).$

However, at most a total flow of 1 can be routed using the edge (n−1,0), so the combined flow on the other n−1 edges must be at least

$\left( {\sum\limits_{0 \leq a < b \leq {n - 1}}{\left( {b - a} \right)\left( {d_{a,b} + d_{b,a}} \right)}} \right) - \left( {n - 1} \right)$

The total flow on those n−1 edges must be less than their totalcapacity, so we obtain that

${\left. {\sum\limits_{0 \leq a < b \leq {n - 1}}{\left( {b - a} \right)\left( {d_{a,b} + d_{b,a}} \right)}} \right) - \left( {n - 1} \right)} \leq {n - 1.}$

The utilization on edge (0,n−1) of our oblivious routing for thesedemands is

${\sum\limits_{0 \leq a < b \leq {n - 1}}\;{\left( {1 - x_{b^{-}a}} \right)\left( {d_{a,b} + d_{b,a}} \right)}} = {{{\sum\limits_{0 \leq a < b \leq {n - 1}}{\left( {b - a} \right)/{n\left( {d_{a,b} + d_{b,a}} \right)}}} \leq {2{\left( {n - 1} \right)/n}}} = {2 - {2\text{/}{n.}}}}$

For the clique topology, the shortest-path routing, where the flow ofeach demand is routed on the direct edge, performs very poorly, withperformance ratio of n−1. We shall see that the optimal obliviousrouting for the clique topology utilizes 2-hop paths.

Lemma 7.3: The optimum oblivious ratio for K_(n) the complete graph on nvertices) is 2-2/n.

Proof: We first show that the ratio is at least 2-2/n: From symmetry andflow conservation, we know that there is an optimal oblivious routingwith the following form: for some χ≧0f _(a,b)(a,b)=χ

-   -   for all distinct a,b

f_(a, b)(a, c) = f_(a, b)(c, b) = (1 − χ)/(n − 2)

-   -   for all distinct a,b,c        f _(a,b)(c,d)=0    -   for all other a,b,c,d

The minimum s-t cut between any two nodes is n−1. Thus, for any given ODpair a, b, a demand a→b of size (n−1) can be routed such that themaximum flow on any edge is 1. By considering such single OD pairdemands a→b, and the edge (a, b), we obtain that the optimal ratio is atleast(n−1)*f _(a,b)(a,b)=(n−1)*χ

We now consider a TM such that there is a demand of 1 for each OD pairc→d (c<d). Such TM can also be routed within unit capacities by routingeach demand c→d on the “direct” edge (c,d). By considering the flow ofour routing on the edge a→b, we have that the optimal ratio is at least

$\left. {{{1^{*}{f_{a,b}\left( {a,b} \right)}} + {\sum\limits_{c \notin {\{{a,b}\}}}{f_{a,b}\left( {a,b} \right)}} + {\sum\limits_{c \notin {\{{a,b}\}}}{f_{c,b}\left( {a,b} \right)}}} = {\chi + {2*\left( {n - 2} \right)*{\left( {1 - \chi} \right)/n}} - 2}} \right).$

from the above two constraints we obtain that the optimal ratio is atleast max{x*(n−1), 2-x}≧2-2/n (the maximum is minimized when x=2/n).

It remains to show that the optimal ratio is at most 2-2/n. We will usea routing of the form (6) with x+2/n, and show that its obliviousperformance ratio is at most 2-2/n. Consider a particular edge a→b.Since a and b have degree n−1, any TM which can be routed with at mostone unit of flow on each edge must satisfy

$\mspace{25mu}{{{\sum\limits_{c \neq a}d_{a,c}} + d_{c,a}} \leq {n - 1}}$$\mspace{25mu}{{{\sum\limits_{c \neq b}d_{b,c}} + d_{c,b}} \leq {n - 1}}$${Therefore},{{{2d_{a,b}} + {2d_{b,a}} + {\sum\limits_{c \notin {\{{a,b}\}}}\left( {d_{a,c} + d_{c,a} + d_{b,c} + d_{c,b}} \right)}} \leq {2\left( {n - 1} \right)}}$

From (6), some optimal oblivious routing then satisfies that the flow onthe edge (a,b) is equal to

$\left. {{\sum\limits_{c,d}\;\left( {{d_{c,d}{f_{c,d}\left( {a,b} \right)}} + {d_{c,d}{f_{c,d}\left( {b,a} \right)}}} \right)} = {{\sum\limits_{{c \notin a},b}{{d_{a,c}\left( {1 - \chi} \right)}/\left( {n - 2} \right)}} + {{d_{b,c}\left( {1 - \chi} \right)}/\left( {n - 2} \right)} + {{d_{c,a}\left( {1 - \chi} \right)}/\left( {n - 2} \right)} + {{d_{c,b}\left( {1 - \chi} \right)}/\left( {n - 2} \right)}}} \right) + {d_{a,b}\chi} + {d_{b,a}\chi}$

By substituting χ=2/n we obtain that the flow on (a,b) is

${{{\sum\limits_{{c \notin a},b}\;{\left( {d_{a,c} + d_{b,c} + d_{c,a} + d_{c,b}} \right)/n}} + {\left( {{2d_{a,b}} + {2d_{b,a}}} \right)/n}} \leq {2{\left( {n - 1} \right)/n}}} = {2 - {2\text{/}n}}$

is the best performance ratio guarantee attainable barring any knowledgeof traffic demands? Our method provides a computation of an optimaloblivious routing and its performance ratio. The results are that theoblivious performance ratio on evaluated topologies range from1.425-1.972, which means that these networks have a routing that on anyTM is guaranteed to have maximum link utilization that is at most43%-97% larger than the best possible specific routing for this TM.While a 43%-97% (worst case) overhead in max utilization is far fromnegligible, the good news is that such guarantees can be obtained withno knowledge whatsoever on the traffic demands. Typically, however, eventhough an exact current estimate of the TM is very hard to obtain, muchabout the TM is known. The TM can vary within some known range or can beestimated to within some known accuracy. In this case, we would like aperformance guarantee with respect to all TMs that lie within somerange. The next question we examine is the sensitivity of theperformance ratio to the error margin within which the TM is known.

Another benefit of systems that are in accord with the principles of thepresent invention is that they perform well under random link failures:The inventors simulated link failures where a random link loses 50% or80% of its capacity. The optimal routing for the non-failed network,although not optimized for failures, typically obtains reasonableperformance under such failures. The explanation is that a goodoblivious routing tends to spread over many available paths for each ODpair, a property that increases its resilience to failures.

From the foregoing it should be obvious that traffic demands areillusive. They are hard to estimate and are dynamic in nature. Goodsystem engineering should seek a routing that performs wellindependently of traffic demands (or for a wide range of demands). Wehave developed methods and systems for obtaining such a routing andarrive at perhaps unexpected results. First, it is possible to obtain asurprisingly good routings with poor or no knowledge of the trafficdemands. On current ISP topologies, we have shown that there exists arouting that guarantees a performance ratio that is less than 2 on anypossible traffic matrix. This demand oblivious routing can be obtainedwithout knowledge of the traffic matrix and by taking only the topology(along with link capacities) into account. With a very limited knowledgeof the TM we can do much better, often obtaining a routing withperformance ratio that is very close to 1 even for error margins of50%-100% in knowledge of the traffic demands.

Similarly, one can obtain a fixed routing that would perform well on anexpected range of demands, thus, reducing the need for routingadjustments when traffic demands shift. Second, it is unlikely that sucha robust routing can be obtained via standard previously-existing tools:The design of a routing with optimal or nearly optimal performanceguarantees seems to require the algorithmic tools we employed here. Weobserved that some natural candidates for routings can perform badly ondemands they are not specifically designed to handle. Moreover, even arouting designed to be optimal on specific TMs deteriorates quickly withthe margins within which the cumulative fraction Max utilization within25% of optimal actual demands deviate from the presumed ones.

FIG. 5 depicts a high level block diagram of a node 500 that is inaccord with the principles of the present invention. The node 500represents a server, e.g. an Asynchronous Transfer Mode (ATM) server,which can be used in a network such as that shown in FIG. 1. The node500 comprises a processor 510 as well as computer readable medium 520for storing control programs 525 and data structures 527. The controlprograms 525 configure the path-based routings of the node 500 in accordwith an optimum network routing as described above.

The processor 510 cooperates with conventional support circuitry 530such as power supplies, clock circuits, cache memory and the like aswell as circuits that assist in executing the control programs 525 inthe memory 520. As such, it is contemplated that some of the processsteps discussed herein as software processes may be implemented withinhardware, for example, as circuitry that cooperates with the processor510 to perform various steps. The node 500 also contains input-outputcircuitry 540 that forms an interface with other nodes in a network.

Although the node 500 is depicted as having a processor 510 thatoperates according to a software program, the invention can beimplemented in hardware, for example, as an application specifiedintegrated circuit (ASIC). As such, the process steps described hereinare intended to be broadly interpreted as being equivalently performedby software, hardware, or a combination thereof. Furthermore, thecomputer readable media 520 is a computer readable memory that storesinformation that can be run and/or accessed by the processor 510.

While the foregoing is directed to embodiments of the present invention,other and further embodiments of the invention may be devised withoutdeparting from the basic scope thereof, and the scope thereof isdetermined by the claims that follow.

1. A method of configuring a network, comprising: obtaining informationabout a plurality of nodes and a plurality of links in the network;identifying possible origin-destination pairs; computing an optimumoblivious ratio of the network, wherein the optimum oblivious ratio isan oblivious ratio of a routing having a minimum oblivious ratio,wherein the oblivious ratio is a worst performance ratio the routing canobtain with respect to any traffic matrix of the network, wherein thecomputing the optimum oblivious ratio is performed by solving a linearprogram, wherein the linear program is based on a reduced set of inputtopologies, wherein degree-one nodes are removed, wherein the optimumoblivious ratio is computed by partitioning the network to 2-edgeconnected components and taking a maximum of the optimum oblivious ratioover the 2-edge connected components; and configuring the network inaccordance with the optimum oblivious ratio.
 2. The method of claim 1,wherein the optimum oblivious ratio is computed using linear constraintson origin-destination pair demands.
 3. The method of claim 1, whereinthe input topologies where a path diversity is not possible are removed.4. A method of configuring a network, comprising: obtaining informationabout a plurality of nodes and a plurality of links in the network;identifying possible origin-destination pairs; computing an optimumnetwork routing, wherein the computing the optimum network routing isperformed by solving a linear program, wherein the linear program isbased on a reduced set of input topologies, wherein degree-one nodes areremoved, wherein the optimum network routing is computed by partitioningthe network to 2-edge connected components and taking a maximum of anoptimum oblivious ratio over the 2-edge connected components, whereinthe optimum oblivious ratio is an oblivious ratio of a routing having aminimum oblivious ratio, wherein the oblivious ratio is a worstperformance ratio the routing can obtain with respect to any trafficmatrix of the network; and configuring the network in accordance withthe optimum network routing.
 5. The method of claim 4, wherein theoptimum network routing is computed using linear constraints onorigin-destination pair demands.
 6. The method of claim 4, wherein theinput topologies where a path diversity is not possible are removed. 7.A network comprising: a plurality of routers that support path-basedrouting; and a plurality of links that are connected to the plurality ofrouters, wherein each path-based routing is configured in accordancewith an oblivious routing configuration based on the plurality ofrouters and the plurality of links, wherein the oblivious routingconfiguration is derived by: identifying possible origin-destinationpairs; computing an optimum network routing based on linear constraintsplaced on origin-destination pair demands, wherein the computing theoptimum network routing is performed by solving a linear program,wherein the linear program is based on a reduced set of input topologiesand wherein degree-one nodes are removed; and configuring eachpath-based routing in accordance with the optimum network routing,wherein the oblivious routing configuration is further derived bypartitioning the network to 2-edge connected components and taking amaximum of an optimum oblivious ratio over the 2-edge connectedcomponents, wherein the optimum oblivious ratio is an oblivious ratio ofa routing having a minimum oblivious ratio, wherein the oblivious ratiois a worst performance ratio the routing can obtain with respect to anytraffic matrix of the network.
 8. The network of claim 7, wherein theinput topologies where a path diversity is not possible are removed. 9.A network, comprising: a plurality of network nodes, each of which hasconfigurable routing characteristics; means for obtaining informationabout the network; means for identifying possible origin-destinationpairs of network nodes; means for computing an optimum oblivious ratioof the network, wherein the optimum oblivious ratio is an obliviousratio of a routing having a minimum oblivious ratio, wherein theoblivious ratio is a worst performance ratio the routing can obtain withrespect to any traffic matrix of the network, wherein the means forcomputing computes the optimum oblivious ratio by solving a linearprogram, wherein the linear program is based on a reduced set of inputtopologies and wherein degree-one nodes are removed, and wherein themeans for computing computes the optimum oblivious ratio by partitioningthe network to 2-edge connected components and taking a maximum of theoptimum oblivious ratio over the 2-edge connected components; and meansfor configuring the plurality of network nodes in accordance with theoptimum oblivious ratio.
 10. The network of claim 9, wherein the meansfor computing computes the optimum oblivious ratio by using linearconstraints on origin-destination pair demands.